1. Field of the Invention
This invention relates to methods and systems for determining molecular structures using x-ray crystallography.
2. Description of the Related Art
In x-ray diffraction crystallography, a crystalline form of the molecule under study is exposed to a beam of x-rays, and the intensity of diffracted radiation at a variety of angles from the angle of incidence is measured. The beam of x-rays is diffracted into a plurality of diffraction xe2x80x9creflections,xe2x80x9d each reflection corresponding to a reciprocal lattice vector. From the diffraction intensities of the reflections, the magnitudes of a series of numbers, known as xe2x80x9cstructure factors,xe2x80x9d are determined. The structure factors in general are complex numbers, having two components in the complex plane, a magnitude and a phase, and are defined by the electron distribution within the unit cell of the crystal.
The structure factors used to calculate atomic coordinates from measured x-ray diffraction intensities are oscillatory functions of the indices of the reciprocal lattice vectors with an overall decaying envelope. One expression for these structure factors has the following form:             Equation  1:        ⁢          xe2x80x83                  xe2x80x83        ⁢                  F        hkl            =              
            ⁢                        1          V                ⁢                              ∑            j                    ⁢                                    q              J                        ⁢                          T              j                        ⁢                          f              j                        ⁢                          {                                                                    cos                    ⁡                                          [                                              2                        ⁢                        π                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                    hx                              j                                                        +                                                          ky                              j                                                        +                                                          lz                              j                                                                                )                                                                    ]                                                        +                                      i                    ⁢                                          xe2x80x83                                        ⁢                                          sin                      ⁡                                              [                                                  2                          ⁢                                                      π                            ⁢                                                          (                                                                                                hx                                  j                                                                +                                                                  ky                                  j                                                                +                                                                  lz                                  j                                                                                            )                                                                                                      ]                                                                                            ,                            ⁢                              xe2x80x83                                                        
where Fhkl is the structure factor for the reciprocal lattice vector with indices h, k, l; qj are the occupancy populations of each site; Tj are the temperature factors which correspond to thermal motions; and fj are the atomic scattering factors. While the populations qj are constants, the temperature factors Tj and atomic scattering factors fj decrease as the indices h, k, l increase.
Working from the magnitudes and phases of the structure factors, the electron density and/or atomic positions within the unit cell of the crystal can be determined. Structural determinations using x-ray diffraction data are described in An Introduction to X-Ray Crystallography by Michael M. Woolfson, Cambridge University Press (1970, 1997), which is hereby incorporated by reference in its entirety.
In principle, all of the x-ray diffraction reflections are capable of being known or measured (i.e., cognizable). However, due to various aspects of the systems used to experimentally measure the reflection intensities, the set of measured intensities may be incomplete, or may contain errors. First, some x-ray diffraction measurement systems do not provide a measurement of the (0, 0, 0) reflection, which can contain useful information regarding the contents of the crystal. Second, the range of reflections accessible by the x-ray measurement system can be constrained to some value, preventing the measurement of reflections corresponding to larger reciprocal lattice vectors. These larger reciprocal lattice vectors can contain high-resolution information (i.e., corresponding to shorter distances in direct space) regarding the crystal structure. Third, various other reflections may be partially or wholly occluded by various portions of the x-ray diffraction measurement system. Fourth, there may be other experimental factors, such as signal-to-noise, which reduce the confidence of a particular measurement by the x-ray measurement system.
According to one aspect of the present invention, a method uses linear prediction analysis to define a first structure factor component for a first reflection from x-ray crystallography data. The x-ray crystallography data comprises a set of cognizable reflections. The method comprises expressing the first structure factor component as a first linear equation in which the first structure factor component is equal to a sum of a first plurality of terms. Each term comprises a product of (1) a structure factor component for a cognizable reflection from the x-ray crystallography data, wherein the cognizable reflection has a separation in reciprocal space from the first reflection, and (2) a linear prediction coefficient corresponding to the separation between the cognizable reflection and the first reflection. The method further comprises calculating values for the linear prediction coefficients. The method further comprises substituting the values for the linear prediction coefficients into the first linear equation, thereby defining the first structure factor component for the first reflection.
According to another aspect of the present invention, a method refines x-ray diffraction data. The method comprises deriving a value of a first structure factor from a linear combination of other structure factors.
According to another aspect of the present invention, a computer readable medium has instructions stored thereon which cause a general purpose computer to perform a method of using linear prediction analysis to define a first structure factor component for a first reflection from x-ray crystallography data. The x-ray crystallography data comprises a set of cognizable reflections. The method comprises expressing the first structure factor component as a first linear equation in which the first structure factor component is equal to a sum of a first plurality of terms. Each term comprises a product of (1) a structure factor component for a cognizable reflection from the x-ray crystallography data, wherein the cognizable reflection has a separation in reciprocal space from the first reflection, and (2) a linear prediction coefficient corresponding to the separation between the cognizable reflection and the first reflection. The method further comprises calculating values for the linear prediction coefficients. The method further comprises substituting the values for the linear prediction coefficients into the first linear equation, thereby defining the first structure factor component for the first reflection.
According to another aspect of the present invention, a computer-implemented x-ray crystallography analysis system comprises a structure factor component generator for generating a first structure factor component for a first reflection from x-ray crystallography data using linear prediction analysis. The x-ray crystallography data comprises a set of cognizable reflections. The first structure factor component is expressed as a first linear equation in which the first structure factor component is equal to a sum of a first plurality of terms. Each term comprises a product of (1) a structure factor component for a cognizable reflection from the x-ray crystallography data, wherein the cognizable reflection has a separation in reciprocal space from the first reflection, and (2) a linear prediction coefficient corresponding to the separation between the cognizable reflection and the first reflection. The system further comprises a calculating module for calculating values for the linear prediction coefficients. The system further comprises a resultant structure factor component definer for defining the first structure factor component for the first reflection by substituting the values for the linear prediction coefficients into the first linear equation.
According to another aspect of the present invention, a computer-implemented x-ray crystallography analysis system comprises a means for generating a first structure factor component for a first reflection from x-ray crystallography data using linear prediction analysis. The x-ray crystallography data comprises a set of cognizable reflections. The first structure factor component is expressed as a first linear equation in which the first structure factor component is equal to a sum of a first plurality of terms. Each term comprises a product of (1) a structure factor component for a cognizable reflection from the x-ray crystallography data, wherein the cognizable reflection has a separation in reciprocal space from the first reflection, and (2) a linear prediction coefficient corresponding to the separation between the cognizable reflection and the first reflection. The system further comprises a means for calculating values for the linear prediction coefficients. The system further comprises a means for defining the first structure factor component for the first reflection by substituting the values for the linear prediction coefficients into the first linear equation.